G'day Tom,
I may be misunderstanding your intentions, but if you want to weight
your data differentially, you can simply incorporate this into your
subroutines which return the residuals vector and jacobian matrix to
the minimizer -- "sigma" in the example from the manual, from memory.
That is, the data structure which you pass into the minimizer is a
vector of x data values, a vector of y data values, and a vector of
errors. The objective function to be minimized is the residuals vector
with elements i = [ ( y_i - f(x_i,a) ) / sigma_i ], and the partial
derivatives include this attenuating factor as well. Meanwhile, the
covariance matrix (co-dependence of your optimized parameters in the
fit) per se is one of the outputs from the minimization routine.
Am I off the mark?
Regards,
john
On 17/08/2009, at 7:57 PM, Tom Banwell wrote:
Hi,
I'm a computer scientist, not a mathematician and had a question
about the Levenberg Marquardt (LM) algorithm and hoped someone would
be able to help or provide some advice. I wanted to perform
weighted Least-Squares and I have the following equation:
[ yi - f(xi,a)]^T Vi [ yi - f(xi,a)]
where yi is the dependent variable, xi is the independent variable,
a are the model parameters to be estimated, and Vi is a covariance
matrix.
I have solved the problem using an unweighted Least-squares but
would prefer to use weighted as some of my data have larger relative
uncertainties. I have seen on the GSL reference manual that I can
perform weighted Least-Squares using a scalar, but I wanted to use
the full covariance matrix, Vi. I did think about using the trace
or determinant of Vi, but not sure if that is mathematically as
sound so wanted to use Vi. My problem is that I can't work out how
to extend my code to include the matrix weight (rather than the
scalar weight) and wondered if I needed to modify the internal gsl
LM algorithm (or maybe rewrite the algorithm myself) or can apply I
apply a matrix weight using the existing gsl LM.
I hope someone can help with this, my email is: tombanwell * at *
hotmail * dot * com
Thanks
Tom
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